# Imaginary-quadratic elliptic curve modularity adapter

Family 030: Modularity of elliptic curves over imaginary quadratic fields. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Arithmetic software teams could connect elliptic-curve data with matching automorphic local parameters across imaginary quadratic fields.

## Applicability and commercial boundary

Modularity existence is not an efficient construction of forms, point counting or a new elliptic-curve security guarantee.

## Initial business decision

Research infrastructure or conditional engineering until a computable implementation, effective bounds and a recurring buyer need are demonstrated. No commercial novelty or profitability is established.

## Next verification action

Identify a finite local-parameter comparison workflow and effective algorithms.

## Evidence scope

The catalog statement was individually reviewed. Manuscript proof and selected formal scope comparison remain queued. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Modularity of elliptic curves over imaginary quadratic fields](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026/paper.pdf).